Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS

2011/03/28 by Zihua Guo, Guo, Zihua, Soonsik Kwon +3 · 4 citations
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.1103.5271

openalex publication_date 2011/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L²(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L²(T) with initial data in L²(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s ≥ 1/6.

Citations

• Normal forms and quadratic nonlinear Klein‐Gordon equations

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• Improved refined bilinear estimates and well-posedness for generalized KdV type equations on ℝ
• Sharp unconditional well-posedness of the 2-d periodic cubic hyperbolic nonlinear Schrödinger equation
• Nonsqueezing property of the coupled KdV type system without Miura transform

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